The Grüneisen parameter, γ, named after Eduard Grüneisen, describes the effect that changing the volume of a crystal lattice has on its vibrational properties, and, as a consequence, the effect that changing temperature has on the size or dynamics of the crystal lattice. The term is usually reserved to describe the single thermodynamic property, which is a weighted average of the many separate parameters entering Grüneisen's original formulation in terms of the phonon nonlinearities.
Thermodynamic definitions
Because of the equivalences between many properties and derivatives within thermodynamics, there are many formulations of the Grüneisen parameter which are equally valid, leading to numerous distinct yet correct interpretations of its meaning. Some formulations for the Grüneisen parameter include: where is volume, and are the principal heat capacities at constant pressure and volume, is energy, is entropy, is the volume thermal expansion coefficient, and are the adiabatic and isothermal bulk moduli, is the speed of sound in the medium, and is density. The Grüneisen parameter is dimensionless.
Grüneisen constant for perfect crystals with pair interactions
The expression for the Grüneisen constant of a perfect crystal with pair interactions in -dimensional space has the form: where is the interatomic potential, is the equilibrium distance, is the space dimensionality. Relations between the Grüneisen constant and parameters of Lennard-Jones, Morse, and Mie potentials are presented in the table below.
Lattice
Dimensionality
Lennard-Jones potential
Mie Potential
Morse potential
Chain
Triangular lattice
FCC, BCC
"Hyperlattice"
General formula
The expression for the Grüneisen constant of a 1D chain with Mie potential exactly coincides with the results of MacDonald and Roy. Using the relation between the Grüneisen parameter and interatomic potential one can derive the simple necessary and sufficient condition for Negative Thermal Expansion in perfect crystals with pair interactions A proper description of the Grüneisen parameter represents a stringent test for any type of interatomic potential.
Microscopic definition via the phonon frequencies
The physical meaning of the parameter can also be extended by combining thermodynamics with a reasonable microphysicsmodel for the vibrating atoms within a crystal. When the restoring force acting on an atom displaced from its equilibrium position is linear in the atom's displacement, the frequencies ωi of individual phonons do not depend on the volume of the crystal or on the presence of other phonons, and the thermal expansion is zero. When the restoring force is non-linear in the displacement, the phonon frequencies ωi change with the volume. The Grüneisen parameter of an individual vibrational mode can then be defined as the logarithmic derivative of the corresponding frequency :
Relationship between microscopic and thermodynamic models
Using the quasi-harmonic approximation for atomic vibrations, the macroscopic Grüneisen parameter can be related to the description of how the vibrational frequencies within a crystal are altered with changing volume. For example, one can show that if one defines as the weighted average where 's are the partial vibrational mode contributions to the heat capacity, such that
Proof
To prove this relation, it is easiest to introduce the heat capacity per particle ; so one can write This way, it suffices to prove Left-hand side : Right-hand side : Furthermore : Thus This derivative is straightforward to determine in the quasi-harmonic approximation, as only the are V-dependent. This yields
